-
Notifications
You must be signed in to change notification settings - Fork 28
Stochastic sampling methodology
SANDY can work as a nuclear data sampling tool applicable to sensitivity analysis (SA) and uncertainty quantification (UQ) problems. The code exploits the basic theory of stochastic sampling to propagate nuclear data covariances through the nuclear models under study.
Nuclear parameters ENDF-6
nuclear data libraries as best-estimate or evaluated data
where
UQ studies aim at propagating the model input uncertainties and/or covariances to the model responses.
Along this working line, the ENDF-6 format allowed storing the covariance matrices
A covariance matrix
From the variance term of a parameter
The off-diagonal terms contain the covariances, a generalization of the variance integral to two different variables
The covariance is a measure of the joint variability of two variables to the
same source of error.
A covariance
The Pearson correlation coefficient is a parameter that varies between -1 and 1 and that can be interpreted as it follows:
-
$\rho_{i,j}=1$ implies that$x_i$ linearly increases as$x_j$ increases; -
$\rho_{i,j}=-1$ implies that$x_i$ linearly increases as$x_j$ decreases; -
$\rho_{i,j}=0$ implies that there is no linear correlation between the variables.
A last important concept, which is often used in the rest of the manual, is the relative covariance.
It is common to refer with the term stochastic sampling to computational approaches that rely on a repeated random sampling of chosen parameters to obtain statistical outcomes of selected responses.
Monte Carlo sampling methods are based on drawing sets of samples from the model input joint probability density function (PDF), as
Given input parameters
One can prove that for any given parameter
$$\overline{x}i \equiv \frac{1}{n}\sum{k=1}^n x_i^{(k)} \rightarrow E\left[x_i\right]$$
Analogously, as
Now, let us assume that the user is interested in a model of the type
The average - or best estimate - and uncertainty of the response are simply
The Monte Carlo sampling approach for uncertainty quantification uses a straightforward approach to reproduce the integrals above.
For any given set
We often refer to
$y^{(k)}$ as to a perturbed response, since it reflects the variation of$y$ generated by the perturbations on$\mathbf{x}$ introduced by the stochastic sampling.
Then, the
Notice that this matrix contains only one row, since so far we only considered models with a single response value.
The observable best estimate and uncertainty can be approximated with
which converge to the real solutions for
Should the model response be a vector
Any two values
In short, the stochastic sampling for uncertainty propagation simply consists in running the same model multiple times, every time replacing the input parameters with a set of samples. This approach brings numerous advantages compared to other uncertainty propagation techniques, such as perturbation theory:
-
its application to any model is straightforward, as it does not interact with the model itself but only with the model inputs;
-
there is no need for developing complicated algorithms to calculate adjoint functions for the model and responses under study;
-
it does not apply only locally around the input best estimates, but it can cover the whole input and output phase space;
-
it can represent any-order effect of the model, as well as interactions between model inputs.
The major drawback of this method lies in the large amount of sets of samples
that must be drawn in order to grant the convergence of the response statistics,
that is, to reduce the statistical error :math:\epsilon
on the response best estimate.
The central limit theorem [ref] tells us that this error is proportional to
The huge improvements of computer performances in the last decades, combined with model simplifications and dimensionality reductions help reduce the computational time of the solvers, thus making Monte Carlo sampling a practical option for nuclear data uncertainty propagation.